Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds

Let \(M\) be a noncompact complete Riemannian manifold. In this paper, we consider the following nonlinear parabolic equation on \(M\) $$\begin{aligned} u_t(x,t)=\Delta u(x,t) + a u(x,t)\ln u(x,t) + bu^{\alpha }(x,t). \end{aligned}$$ We prove a Li–Yau type gradient estimate for positive solutions to the above equation; as an application, we also derive the corresponding Harnack inequality. These results generalize the corresponding ones proved by Li (J Funct Anal 100:233–256, 1991).